Well, I wouldn't use the language "identified this as a square matrix by column matrix." Sakurai appears to have identified Eq. 1.3.25 as an application of the rule for multiplication of a column vector by a square matrix. The final result, as we can see, is a bracket (a scalar). To go from the first line of 1.3.25 to the second, all Sakurai did was insert the identity in the form of

That's certainly a square matrix.

I guess I'm not entirely sure what your question is. Perhaps my comments could help to clarify your question? Are you asking why Sakurai can write down Eq. 1.3.25? Or are you asking how he recognized 1.3.25 as, partly, the multiplication of a square matrix with a column vector? Or something else?

Or are you asking how he recognized 1.3.25 as, partly, the multiplication of a square matrix with a column vector? Or something else?

Apologies for lack of clarity...yes this what I am looking for...additionally how he arrives on the next line 1.3.26? Ie how the kets are column matrices and the bra as row matrices? I am just looking for that link...

Typically, kets are defined as column vectors. (Incidentally, in your last post, you use the phrases "column vector" and "column matrix". Are you meaning the same thing by those two terms?) Bras are something for which you have to deduce their structure such that the inner product makes sense - they are vectors in the dual space.

Typically, kets are defined as column vectors. (Incidentally, in your last post, you use the phrases "column vector" and "column matrix". Are you meaning the same thing by those two terms?) Bras are something for which you have to deduce their structure such that the inner product makes sense - they are vectors in the dual space.

Ok, that is good information. I also see that an inner product is a complex number therefore in matrix representation this has to be in the form